Problem 43 - Algebra 2

Question

MUSIC When represented on oscilloscope, the note A above middle C has a period of \(\frac{1}{440} .\) Which of the following can be an equation for an oscilloscope graph of this note? The amplitude of the graph is \(K\) . $$ \text { a. }y=K \sin 220 \pi t \quad \text { b. } y=K \sin 440 \pi t \quad \text { c. } y=K \sin 880 \pi t $$

Step-by-Step Solution

Answer

The correct equation is y = K \sin(880\pi t).

Step 1: Understand the Period of the Wave

The period of a wave is the time it takes to complete one full cycle. Given that the period is \( \frac{1}{440} \), this represents the fundamental frequency of the note A above middle C.

Step 2: Understand the Relation Between Period and Frequency in a Wave Equation

The general equation for a sine wave is \( y = K \sin(2\pi ft) \), where \( f \) is the frequency. The period \( T \) is related to the frequency by \( T = \frac{1}{f} \), or equivalently, \( f = \frac{1}{T} \). Here, \( T = \frac{1}{440} \), so \( f = 440 \).

Step 3: Identify the Appropriate Coefficient in the Sine Function

In the sine wave equation, \( y = K \sin(2\pi ft) \), substituting \( f = 440 \) gives \( y = K \sin(2\pi \times 440 \times t) \). Simplifying gives \( y = K \sin(880\pi t) \).

Step 4: Compare with Given Options

Compare the derived equation \( y = K \sin(880\pi t) \) with the options provided. Option c, \( y = K \sin(880 \pi t) \), matches the derived equation, indicating it as the correct solution.

Key Concepts

Sine Wave

A sine wave is a smooth, periodic oscillation that is typically used to represent continuous waveforms in various fields such as physics and engineering. When graphed, a sine wave looks like a wave that gracefully rises and falls, creating a repetitive pattern over time. This shape is pivotal for understanding natural phenomena like sound waves.

Sine waves are defined by three key characteristics:

• Amplitude: The maximum distance from the wave's equilibrium point. It represents how "strong" or "loud" a wave is if you're thinking of sound.
• Frequency: The number of cycles the wave completes in a second. Higher frequency means more cycles, thus a "faster" wave.
• Period: The time it takes for the wave to complete one full cycle.

These elements are crucial in music, with sine waves helping to describe the pitch and tone of notes, such as the note A above middle C.

Frequency and Period

In wave mechanics, frequency and period are intimately connected concepts that describe a wave's temporal properties.

The frequency of a wave, denoted by the letter \( f \), is the number of complete waves or cycles that pass a point in one second. It is measured in Hertz (Hz). High-frequency waves have many cycles in a short time frame, making them appear "tight" when visualized.

The period of a wave, symbolized by \( T \), is the time taken for one complete cycle of the wave. It is inversely related to frequency – as one increases, the other decreases – with the relationship given by:

• The formula \( T = \frac{1}{f} \), where \( T \) is the period in seconds and \( f \) is the frequency in Hertz.

For example, the note A above middle C has a frequency of 440 Hz, which means it has a period of \( \frac{1}{440} \) seconds. This ability to mathematically relate period and frequency allows us to precisely describe musical notes and understand their waveforms.

Wave Equation

The wave equation is a quintessential formula used in physics and engineering to describe the motion and properties of waves. For a basic sine wave, the wave equation is written as:

• \( y = K \sin(2\pi ft) \)

In this equation:

• \( y \) represents the displacement at time \( t \).
• \( K \) is the amplitude, designating the maximum displacement from the central axis.
• \( f \) represents the frequency, dictating how many cycles occur per second.
• \( t \) is the time variable, crucial for determining the wave's position at a given moment.
• \( 2\pi f \) is the angular frequency, ensuring the correct scale for the sinusoidal wave to repeat correctly.

This equation enables us to model waves and predict their behavior under different conditions. In the context of a musical note like A above middle C, which has a frequency of 440 Hz, substituting this into the wave equation provides a specific model of that note's oscillation. Understanding this function helps in visualizing and creating sound waves correctly in instruments and audio technology.